Oscillation of Mertens’ product formula

Harold G. Diamond[1]; Janos Pintz[2]

  • [1] Univ. of Illinois Dept. of Math. 1409 West Green Street Urbana, IL 61801 USA
  • [2] Rényi Mathematical Institute Hungarian Academy of Sciences Reáltanoda u. 13-15 Budapest, H-1053, Hungary

Journal de Théorie des Nombres de Bordeaux (2009)

  • Volume: 21, Issue: 3, page 523-533
  • ISSN: 1246-7405

Abstract

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Mertens’ product formula asserts that ∏ p ≤ x 1 - 1 p log x → e - γ as x → ∞ . Calculation shows that the right side of the formula exceeds the left side for 2 ≤ x ≤ 10 8 . It was suggested by Rosser and Schoenfeld that, by analogy with Littlewood’s result on π ( x ) - li x , this and a complementary inequality might change their sense for sufficiently large values of x . We show this to be the case.

How to cite

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Diamond, Harold G., and Pintz, Janos. "Oscillation of Mertens’ product formula." Journal de Théorie des Nombres de Bordeaux 21.3 (2009): 523-533. <http://eudml.org/doc/10897>.

@article{Diamond2009,
abstract = {Mertens’ product formula asserts that\[ \prod \_\{p \le x\} \Big ( 1 - \frac\{1\}\{p\} \Big )\, \log x \, \rightarrow \, e^\{-\gamma \} \]as $x \rightarrow \infty $. Calculation shows that the right side of the formula exceeds the left side for $2 \le x \le 10^8$. It was suggested by Rosser and Schoenfeld that, by analogy with Littlewood’s result on $\pi (x) - \textrm\{li \} x$, this and a complementary inequality might change their sense for sufficiently large values of $x$. We show this to be the case.},
affiliation = {Univ. of Illinois Dept. of Math. 1409 West Green Street Urbana, IL 61801 USA; Rényi Mathematical Institute Hungarian Academy of Sciences Reáltanoda u. 13-15 Budapest, H-1053, Hungary},
author = {Diamond, Harold G., Pintz, Janos},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Mertens’ product formula; oscillation; Euler’s constant; Riemann hypothesis; zeta function; Mertens' theorem; oscillation theorem; Tauberian theory},
language = {eng},
number = {3},
pages = {523-533},
publisher = {Université Bordeaux 1},
title = {Oscillation of Mertens’ product formula},
url = {http://eudml.org/doc/10897},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Diamond, Harold G.
AU - Pintz, Janos
TI - Oscillation of Mertens’ product formula
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 3
SP - 523
EP - 533
AB - Mertens’ product formula asserts that\[ \prod _{p \le x} \Big ( 1 - \frac{1}{p} \Big )\, \log x \, \rightarrow \, e^{-\gamma } \]as $x \rightarrow \infty $. Calculation shows that the right side of the formula exceeds the left side for $2 \le x \le 10^8$. It was suggested by Rosser and Schoenfeld that, by analogy with Littlewood’s result on $\pi (x) - \textrm{li } x$, this and a complementary inequality might change their sense for sufficiently large values of $x$. We show this to be the case.
LA - eng
KW - Mertens’ product formula; oscillation; Euler’s constant; Riemann hypothesis; zeta function; Mertens' theorem; oscillation theorem; Tauberian theory
UR - http://eudml.org/doc/10897
ER -

References

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  10. J. E. Littlewood, Sur la distribution des nombres premiers. Comptes Rendus Acad. Sci. Paris 158 (1914), 1869–1872. Zbl45.0305.01
  11. F. Mertens, Ein Beitrag zur analytischen Zahlentheorie. J. reine angew. Math. 78 (1874), 46–62. 
  12. H. L. Montgomery and R. C. Vaughan, Multiplicative number theory, I. Classical theory. Cambridge Studies in Adv. Math. 97. Cambridge Univ. Press, 2007. MR2378655. Zbl1142.11001MR2378655
  13. J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers. Illinois J. Math. 6 (1962), 64–94. MR0137689 (25 #1139). Zbl0122.05001MR137689
  14. J. Sondow and E. W. Weisstein, Mertens’ Theorem. MathWorld–A Wolfram Web Resource, http://mathworld.wolfram.com/MertensTheorem.html. 

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